US7418353B2 - Determining film stress from substrate shape using finite element procedures - Google Patents
Determining film stress from substrate shape using finite element procedures Download PDFInfo
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- US7418353B2 US7418353B2 US11/245,938 US24593805A US7418353B2 US 7418353 B2 US7418353 B2 US 7418353B2 US 24593805 A US24593805 A US 24593805A US 7418353 B2 US7418353 B2 US 7418353B2
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01L—MEASURING FORCE, STRESS, TORQUE, WORK, MECHANICAL POWER, MECHANICAL EFFICIENCY, OR FLUID PRESSURE
- G01L5/00—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes
- G01L5/0047—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes measuring forces due to residual stresses
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Abstract
Description
σf =E s t 2 s K/6t f(1−νs) (1)
where,
-
- σf=film normal stress;
- Es=elastic modulus of substrate;
- ts=substrate thickness;
- tf=film thickness;
- νs=Poisson's ratio of the substrate;
- K=curvature caused by intrinsic stress.
and
When a tolerance of the normalized distance is specified (denoted as η), a polynomial order N0 is selected as an initial guess, and the normalized distance between
Δ(
If it is not satisfied, N0 has to be increased and the inequality of Eq. (4) should be checked again. These steps are repeatedly performed until the inequality is satisfied and the necessary polynomial order is identified.
{Fs}=[Ks]{Ds} (5)
-
- where, {Fs}=nodal forces at the top surface of the substrate, six components at each node;
- [Ks]=stiffness matrix of the substrate;
- {Ds}=displacements of the substrate, six components at each node.
The stiffness matrix [Ks] of the substrate includes known values based on the material and geometric characteristics of thesubstrate 14 under investigation.
- where, {Fs}=nodal forces at the top surface of the substrate, six components at each node;
{Ff}=−{Fs}. (6)
[Kf]{Df}={Ff} (7)
-
- where, [Kf]=film stiffness matrix;
- {Df}=displacements of the film.
The film stiffness matrix [Kf] includes known values determined based on the material and geometric characteristics of thefilm 12.
- {Df}=displacements of the film.
- where, [Kf]=film stiffness matrix;
{σ}=[E][b]{d} (8)
-
- where, [E]=elastic matrix (film);
- [b]=strain-displacement matrix (film);
- {d}=displacement matrix of an element (film). The film elastic matrix [E] contains known values based on the physical characteristics of the film.
- where, [E]=elastic matrix (film);
TABLE I |
Material and geometric characteristics of the |
measured EUV reticle. |
Elastic | Side | |||||
modulus | Poisson's | Thickness | length | |||
Parameter | E(GPa) | Ratio v | t(mm) | a(mm) | ||
Substrate | 69.3 | 0.17 | 6.35 | 152.4 | ||
Multilayer | 180 | 0.29 | 2.8 × 10−4 | 152.4 | ||
The out-of-plane displacement caused by intrinsic film stress was measured by an interferometer. The measured contours were not axisymetric. A detailed investigation showed that the two principal curvature radii were neither constant over the whole substrate nor equal to each other at most points, i.e., the reticle shape was not spherical. Thus, Stoney's equation would not produce meaningful stress results and a method of determining film stress from substrate shape using finite element procedures in accordance with the present invention was used to determine the stress in the multilayer thin film reflector.
σa=(σx+σy)/2 (9)
The average stress was selected as a stress control parameter. The target value was 480 MPa compressive and uniform. In most areas the average stress was lower than the targeted values, and exceeded the target value only in the small neighborhoods of the corners of the film.
σr=σc(1−r/r 0) (10)
σθ=σC[1−(1+2νf)r/(2+νf)r 0] (11)
where σc is the maximum stress at the center of the film, r is the radial coordinate, ro is the outer radius and νf is Poisson's ratio of the film.
The radial stress and the circumferential stress have the same maximum value at the center. At the outer edge, the radial stress decreases to zero, but the circumferential stress is not equal to zero. The corresponding out-of-plane displacement (OPD) of the substrate is given by:
where ts and tf are the substrate and film thicknesses, respectively, Es is the elastic modulus of the substrate, and νs is Poisson's ratio of the substrate. The analytical solution is based on elasticity theory and satisfies equilibrium, compatibility and boundary conditions. Thus, it can be used as a benchmark for the accuracy of determining film stress using prior art methods that employ Stoney's equation and using a procedure for determining film stress from substrate shape using finite element procedures in accordance with the present invention. The material and geometric parameters for the exemplary test case are listed in Table II:
TABLE II |
Material and geometric characteristics for the |
numerical test case. |
Elastic modulus | Poisson's | Thickness | Outer radius | |
Parameter | E(GPa) | Ratio v | t(μm) | r0(mm) |
Substrate (Si) | 160 | 0.12 | 625 | 50 |
Film (Cr) | 248 | 0.30 | 0.10 | 50 |
The radial stress, σr, has its maximum value of 500 MPa at the center and decreases linearly to zero at the outer edge. The circumferential film stress, σθ, was not equal to θr except at the center, but it also decreased linearly from the center to the outer edge. The minimum value of σθ was 150 MPa at the outer edge.
Claims (19)
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US61797004P | 2004-10-12 | 2004-10-12 | |
US11/245,938 US7418353B2 (en) | 2004-10-12 | 2005-10-07 | Determining film stress from substrate shape using finite element procedures |
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WO (1) | WO2007013884A2 (en) |
Cited By (4)
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US20070180919A1 (en) * | 2004-06-01 | 2007-08-09 | California Institute Of Technology | Characterizing Curvatures and Stresses in Thin-Film Structures on Substrates having Spatially Non-Uniform Variations |
US7930113B1 (en) * | 2007-04-17 | 2011-04-19 | California Institute Of Technology | Measuring stresses in multi-layer thin film systems with variable film thickness |
US9601391B2 (en) | 2015-03-12 | 2017-03-21 | The United States Of America As Represented By The Administrator Of The National Aeronautics And Space Administration | Mechanical stress measurement during thin-film fabrication |
US9625823B1 (en) * | 2010-06-17 | 2017-04-18 | Kla-Tencor Corporation | Calculation method for local film stress measurements using local film thickness values |
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US10024654B2 (en) | 2015-04-06 | 2018-07-17 | Kla-Tencor Corporation | Method and system for determining in-plane distortions in a substrate |
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- 2005-10-11 WO PCT/US2005/036485 patent/WO2007013884A2/en active Application Filing
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Cited By (6)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20070180919A1 (en) * | 2004-06-01 | 2007-08-09 | California Institute Of Technology | Characterizing Curvatures and Stresses in Thin-Film Structures on Substrates having Spatially Non-Uniform Variations |
US7966135B2 (en) | 2004-06-01 | 2011-06-21 | California Institute Of Technology | Characterizing curvatures and stresses in thin-film structures on substrates having spatially non-uniform variations |
US7930113B1 (en) * | 2007-04-17 | 2011-04-19 | California Institute Of Technology | Measuring stresses in multi-layer thin film systems with variable film thickness |
US9625823B1 (en) * | 2010-06-17 | 2017-04-18 | Kla-Tencor Corporation | Calculation method for local film stress measurements using local film thickness values |
US9601391B2 (en) | 2015-03-12 | 2017-03-21 | The United States Of America As Represented By The Administrator Of The National Aeronautics And Space Administration | Mechanical stress measurement during thin-film fabrication |
US10580706B2 (en) | 2015-03-12 | 2020-03-03 | United States Of America As Represented By The Administrator Of Nasa | Thin-film fabrication system employing mechanical stress measurement |
Also Published As
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WO2007013884A2 (en) | 2007-02-01 |
WO2007013884A3 (en) | 2007-03-15 |
US20060123919A1 (en) | 2006-06-15 |
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